for flow curves without a yield point.
3.3.6.2.1a) Ostwald/de Waele, or power-law: τ = c ⋅ γ ̇ p
Flow curve model function according to W. Ostwald jun. (of 1925 [3.2]) and A. de Waele (of 1923 [3.39]) with “flow coefficient” c [Pas] and exponent p. Sometimes, c is referred to as “consistency”, and p to as “flow index” or “power-law index”. It counts: p = 1 for ideal-viscous flow behavior, p < 1 for shear-thinning, p > 1 for shear-thickening.
A disadvantage of this model function is that for flow curves of most polymer solutions and melts it cannot be fitted as well in the low-shear range (since it was developed for linear scaled diagrams) as well as in the high-shear range. These are the ranges of zero-shear viscosity and infinite-shear viscosity. Despite this, the model function is often used in the polymer industry to be fitted in the medium shear rate range (see Figures 3.6 and 3.7, 3.15 and 3.16, and the curve overview of Chapter 3.3.5).
3.3.6.2.2b) Steiger/Ory: γ ̇ = c1 ⋅ τ + c2 ⋅ τ3
Flow curve model function with the (Steiger/Ory) coefficients c1 [1/Pas] and c2 [1/Pa3 ⋅ s],
(of 1961 [3.40])
3.3.6.2.3c) Eyring/Prandtl/Ree or Ree-Eyring: γ ̇ = c1 ⋅ sinh(τ/c2)
Flow curve model function with (EPR) factor c1 [1/s] and “scaling factor” c2 [Pa],
3.3.6.3Model functions for flow behavior with zero-shear viscosity and infinite-shear viscosity
Listed below are ten model functions showing the following parameters:
Zero-shear viscosity η0 = lim γ ̇ → 0 η( γ ̇ ) and infinite-shear viscosity η∞ = lim γ ̇ → ∞ η( γ ̇ )
These model functions have been designed for uncrosslinked and unfilled polymers and are not suitable for dispersions and gels (see also Chapter 3.3.2.1).
3.3.6.3.1a) Cross:
and simplified:
Viscosity curve model function with Cross constant c [s] and Cross exponent p (of 1965 [3.43]). For the simplified version “Cross 0”, it is assumed that η∞ is very low in contrast to η0 and can therefore be ignored. This is usually the case for all concentrated polymer solutions and melts, since here, the viscosity values usually decrease at least by two decades.
3.3.6.3.2b) Carreau:
or as
simplified:
Viscosity curve model function with Carreau constant c1 [s], Carreau exponent p, slope value c of the viscosity curve at high shear rates on a log/log scale, and the shear rate value γ ̇ c at the bend between the plateau of η0 and the falling η-curve in the range of shear-thinning behavior (of 1968/1972 [3.44]).
For the simplified version “Carreau 0”, the same assumptions are made as for the “Cross 0” model above. The Cross and Carreau model functions are similar and they are often used by people working in R & D in the polymer industry. Modifications of the Carreau model:
1) Carreau/Gahleitner :
Viscosity curve model function with Gahleitner exponent p1. For p1 = 2, the model is identical to the Carreau model (in 1989 [3.45]).
2) Carreau/Yasuda :
Viscosity curve model function with Yasuda exponent p1, relaxation time λ [s], and power-law-
index p. It counts: p = 1 for ideal-viscous flow behavior,
p < 1 for shear-thinning, p > 1 for shear-thickening (of 1981 [3.46] [3.47]).
3.3.6.3.3c) Krieger/Dougherty:
Viscosity curve model function with the shear stress τ = τc at the viscosity value (η0/2), assuming that the value of η∞ is low in comparison to η0 (of 1959 [3.48]).
3.3.6.3.4d) Vinogradov/Malkin:
Viscosity curve model function with the coefficients c1 [s] and c2 [s], and exponent p (of 1980 [3.49]).
3.3.6.3.5e) Ellis and Sisko:
Ellis: τ = η0 ⋅ γ ̇ + c ⋅ γ ̇ pandSisko: τ = c ⋅ γ ̇ p + η∞ ⋅ γ ̇
Flow curve model functions with “consistency” c and “index” p.
These model functions are especially designed to describe the behavior at low shear rates (according to Ellis, in 1927 [3.50]), and at high shear rates (acc. to Sisko, in 1958 [3.51).
3.3.6.3.6f) Exponential or e-function: η( γ ̇ ) = η0 ⋅ exp (–c ⋅ γ ̇ )
Viscosity curve model function showing an exponential shape, with the “factor” c [s].
Note: exp(xyz) means exyz, using Euler’s number e = 2.718...
3.3.6.3.7g) Philipps/Deutsch:
Flow